Avoid unnecessary touching, except for when cleaning.
Do not expose trophies to constant, direct sunlight.
Keep in mind the amount of humidity. When exposed to too much humidity, mold could form. Too low humidity could cause mount to dry and crack with time,
Do not hang mount where it is frequently exposed to heat, like above the fireplace. Constant exposure to heat can cause damage to your mount.
Place your trophies in a safe space in your home, where it will not be accidentally bumped an constantly moved.
Keep children and pets away from your trophies.
Regularly clean trophies with duster or slightly damp cloth. Take rugs outside and shake off. Hang in a cool place for ventilation. Do not wash!
You could also use an air compressor to blow of dust.
Limit traffic on rugs to preserve hair and keep your hide in good condition.